Optimal. Leaf size=161 \[ \frac {(1-m) \text {Int}\left (\frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2},x\right )}{2 d}+\frac {(f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}-\frac {b c \sqrt {1-c^2 x^2} (f x)^{m+2} \, _2F_1\left (\frac {3}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{2 d^2 f^2 (m+2) \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.21, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}+\frac {(b c) \int \frac {(f x)^{1+m}}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2 f}+\frac {(1-m) \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}\\ &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}+\frac {(1-m) \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}+\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\left (-1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2 f \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}+\frac {(1-m) \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2 f \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}-\frac {b c (f x)^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {3}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{2 d^2 f^2 (2+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(1-m) \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}\\ \end {align*}
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Mathematica [A] time = 11.21, size = 0, normalized size = 0.00 \[ \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.25, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x \right )^{m} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}{\left (-c^{2} d \,x^{2}+d \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a \left (f x\right )^{m}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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